For a space-time domain \(\Omega \times J\in [0,1] \times [0,1]\) , let the number of time-steps be \(n_t=20\)  and the spatial intervals be \(n_x=2^n+1\) , where \(n\)  is the number of qubits, and \(\delta _x=1\)  is the diffusion parameter. We employ the Dirichlet boundary condition with boundary values \((g_L,g_R )=(1,0)\)  and initial values \(u_0= \vec{0}\) . With initial random parameters \(\theta ^0 \in [0,2 \pi ]\) , we run a limited-memory Broyden-Fletcher-Goldfarb-Shanno boxed (L-BFGS-B) optimizer [1]}, [2]}, [3]}, [4]} to optimize \(\theta \)  with absolute and gradient tolerances set at \(10^{-8}\) .
